COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Complementary Angles :

If the sum of two angles is 90⁰, then those two angles are called as complementary angles.

Example :

30° and 60° are complementary angles.

Because, 

30° + 60°  =  90°

Clearly, 30° is the complement of 60° and 60° is the complement of 30°.

Supplementary Angles :

If the sum of two angles is 180⁰, then those two angles are called as supplementary angles.

Example :

120° and 60° are supplementary angles.

Because,

120° + 60°  =  180°

Clearly, 120° is the supplement of 60° and 60° is the supplement of 120°.

Practice Questions

Question 1 :

The measure of an angle is 41°. What is the measure of a complementary angle?

Answer :

Let x be the measure of the required complementary angle.

Because x and 41° are complementary angles, 

x + 41°  =  90°

Subtract 41° from each side. 

x = 49°

So, the measure of the complementary angle is 49°. 

Question 2 :

The measure of an angle is 62°. What is the measure of a complementary angle?

Answer :

Let x be the measure of the required complementary angle.

Because x and 62° are complementary angles, 

x + 62°  =  90°

Subtract 62° from each side. 

x  =  28°

So, the measure of the complementary angle is 28°.

Question 3 :

The measure of an angle is 108°. What is the measure of a supplementary angle?

Answer :

Let x be the measure of the required supplementary angle.

Because x and 108° are supplementary angles, 

x + 108°  =  180°

Subtract 108° from each side. 

x  =  72°

So, the measure of the supplementary angle is 72°.

Question 4 :

The measure of an angle is 89°. What is the measure of a supplementary angle?

Answer :

Let x be the measure of the required supplementary angle.

Because x and 41° are supplementary angles, 

x + 89°  =  180°

Subtract 89° from each side. 

x  =  91°

So, the measure of the supplementary angle is 91°.

Question 5 :

Two angles are complementary. If one of the angles is double the other angle, find the two angles. 

Answer :

Let x be one of the angles. 

Then the other angle is 2x.

Because x and 2x are complementary angles, we have

x + 2x  =  90°

3x  =  90

Divide each side by 3. 

x  =  30

And,

2x  =  2(30)  =  60

So, the two angles are 30° and 60°.

Question 6 :

Two angles are complementary. If one angle is two times the sum of other angle and 3, find the two angles.

Answer :

Let x and y be the two angles which are complementary.

So, we have

x + y  =  90° -----> (1)

Given : One angle is two times the sum of other angle and 3. 

Then, 

x  =  2(y + 3)

x  =  2y + 6 ----->(2)

Now, substitute (2y + 6) for x in (1). 

(1)-----> 2y + 6 + y  =  90

3y + 6  =  90

Subtract 6 from each side. 

3y  =  84

Divide each side by 3.

y  =  28

Substitute 28 for y in (2).

(2)-----> x  =  2(28) + 6

x  =  56 + 6

x  =  62

So, the two angles are 62° and 28°.

Question 7 :

Find the value of  x :

Answer :

From the picture above, it is clear that the angles x and 2x are complementary.  

Then,

x + 2x  =  90

Simplify. 

3x  =  90

Divide each side by 3.

x  =  30

So, the value of x is 30.

Question 8 :

Find the value of  x :

Answer :

From the picture above, it is clear that the angles (x+1), (x-1) and (x+3) are complementary.  

Then,

(x+1) + (x-1) + (x+3)  =  90

x + 1 + x - 1 + x + 3  =  90

Simplify.

3x + 3  =  90

Subtract 3 from each side. 

3x  =  87

Divide each side by 3.

x  =  29

So, the value of x is 29.

Question 9 :

Find the value of  x :

Answer :

From the picture above, it is clear that (2x+3) and (x-6) are supplementary angles. 

Then,

(2x+3) + (x-6)  =  180

2x + 3 + x - 6  =  180

Simplify. 

3x - 3  =  180

Add 3 to each side. 

3x  =  183

Divide each side by 3. 

x  =  61

So, the value of x is 61.

Question 10 :

Find the value of  x :

Answer :

From the picture above, it is clear (5x+4), (x-2) and (3x+7) are supplementary angles. 

Then, 

(5x+4) + (x-2) + (3x+7)  =  180

5x + 4 + x -2 + 3x + 7  =  180

Simplify. 

9x + 9  =  180

Subtract 9 from each side.

9x  =  171

Divide each side by 9.

x  =  19

So, the value of x is 19.

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